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Fast Fourier Transform

  • Sujit Kumar BoseEmail author
Chapter
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Part of the Forum for Interdisciplinary Mathematics book series (FFIM)

Abstract

If f is an integrable function of a variable \(t\in (-\ ,\,\infty )\), then its Fourier integral or Fourier transform may be defined as the complex function F of a variable \(\omega \in (-\infty ,\,\infty )\) where \(F(\omega ):=\int _{-\infty }^{\infty }f(t)\,e^{-i\omega t}\, dt,\quad i=\sqrt{-1}\). The most important fact about definition (1) is that if it is viewed as an integral equation, its solution under very general conditions is
$$f(t)=\frac{1}{2\pi }\int _{-\infty }^{\infty }F(\omega )\,e^{i\omega t}\, d\omega (2)$$
Equation (2) is called the inverse transform of the direct transform of (1), and the two constitute the Fourier transform pair.

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.S. N. Bose National Centre for Basic SciencesKolkataIndia

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